Need Web browser for older computer using OS 10.13.6

you should try Firefox Dynasty (GitHub project) - constantly being updated and the latest version is based on Firefox 134. It can be got here (bottom right of page, Releases):

https://github.com/i3roly/firefox-dynasty

or you can get it with this Preference Pane which can always be used to get the latest version:

https://github.com/Wowfunhappy/Firefox-Dynasty-Downloader-PrefPane/releases .

Firefox 115 ESR

Chromium 116 (also Chrome 116, Opera 102, or Brave 1.57)

Maybe Pale Moon if your work doesn't require a mainstream browser.

If none of the above work for your job specific needs then you will need a newer computer, (and generally a computer you use for a job should probably should be new enough to receive security updates anyway).

Have you tried Firefox ESR 115 (since later versions appear to require macOS 10.15 Catalina)?

https://www.mozilla.org/en-US/firefox/all/desktop-esr/osx/en-US/

I thought the same as you until I found that people have created versions of modern browsers (on GitHub) for use on older Macs running older OS's.

See my post above about Chromium Legacy (Chrome 124) and Firefox Dynasty (Firefox 134) - both access the internet fine.

Chromium Legacy - if installed using Chromium Legacy Downloader - will automatically update the ISRG X1 certificate chain.

You may also need to update your ISRG X1 certificate: https://letsencrypt.org/certificates/, by downloading the X1 .pem file, manually adding it to your system keychain and customizing the trust settings once it is installed (most forum answers in various places will say to blanket "trust always", but you really only need to do it for SSL in order to access websites).

You will find it difficult, if not impossible to use a 16 year old Mac on the internet. Might be time to replace it.

FWIW, I have been using Firefox 115.18.0esr (32-bit) with an equally old Windows 7 laptop for many web sites (including this one) with good results.